Two Variants of the Reciprocal Super Catalan Matrix

pISSN 1225-6951 eISSN 0454-8124 c

⃝ Kyungpook Mathematical Journal

Two Variants of the Reciprocal Super Catalan Matrix

Emrah Kılıc¸

Department of Mathematics, TOBB University of Economics and Technology, 06560 Ankara, Turkey

e-mail : ekilic@etu.edu.tr Nes¸e ¨Om¨ur∗

Department of Mathematics, Kocaeli University, 41380 ˙Izmit Kocaeli, Turkey e-mail : neseomur@kocaeli.edu.tr

Sibel Koparal and Y¨ucel T¨urker Ulutas¸

Department of Mathematics, Kocaeli University, 41380 ˙Izmit Kocaeli, Turkey e-mail : sibel.koparal@kocaeli.edu.tr and turkery@kocaeli.edu.tr

Abstract. In this paper, we define two kinds variants of the super Catalan matrix as well as their q-analoques. We give explicit expressions for LU-decompositions of these matrices and their inverses.

1. Introduction

For a given sequence{an}∞_n=0, the Hankel matrix is defined by

    a0 a1 a2 ... a1 a2 a3 ... a2 a3 a4 ... ... ... ... ...     .

One can obtain a combinatorial matrix having interesting properties from a Hankel matrix. For example, the Hilbert matrix Hn = [hij] is defined by hij =

1

i+j−1 (for more details, see [2, 4]) and the Filbert matrixFn = [fij] is defined by

* Corresponding Author.

Received July 27, 2015; accepted December 10, 2015. 2010 Mathematics Subject Classification: 15B05, 05A05.

Key words and phrases: Catalan matrix, q-binomial coefficients, LU-decompositions, de-terminant.

fij = _Fi+j−11 (see [1, 6]). Clearly they are of the forms Hn=     1 1 1 2 1 3 ... 1 2 1 3 1 4 ... 1 3 1 4 1 5 ... ... ... ... ...     and Fn=     1 F0 1 F1 1 F2 ... 1 F1 1 F2 1 F3 ... 1 F2 1 F3 1 F 4 ... ... ... ... ...     , where Fn is nth Fibonacci number.

Throughout this paper, we will use the q-Pochhammer symbol

(x; q)n= (1− x) (1 − xq) ...

(

1− xqn−1) and the Gaussian q-binomial coefficients

[ n k ] q = (q; q)n (q; q)k(q; q)n−k . It is clearly that (1.1) lim q→1 [ n k ] q = ( n k ) ,

where(n_k)is the usual binomial coefficient. The Cauchy binomial theorem is given by

n ∑ k=0 q(k+12 ) [ n k ] q xk= n ∏ k=1 ( 1 + xqk),

and Rothe’s formula (see [2]) is given by

n ∑ k=0 (−1)kq(k2) [ n k ] q xk = (x; q)n = n_∏−1 k=0 ( 1− xqk).

Prodinger [3] consider the reciprocal super Catalan matrix M with the entries mij =

i!(i+j)!j!

(2i)!(2j)! and obtain explicit formulae for its decomposition, the

LU-decomposition of its inverse, and some related matrices. For all results, q-analogues are also presented.

We rewrite the matrix M in terms of three binomial coefficients, two of them is in the reciprocal form, as shown

mij = i!(i + j)!j! (2i)!(2j)! = i!j! (2i)!(2j)!= ( 2i i )₋₁( 2j j )₋₁( i + j i ) .

By inspiring the matrix M, we consider two kinds variants of it by keeping only one binomial coefficient in reciprocal form in the first one and two binomial

coefficients in reciprocal forms in the second one. Also we will add two additional parameters to each one. Now we define these matrices: The first one is the matrix A = [aij] of order n with the entries

aij= ( 2i + m i )( 2j + t j )−1(_{i + j} i )

and the second one is the n× n matrix B with the entries

bij = ( 2i + m i )−1(_{2j + t} j )( i + j i )−1 ,

where m and t are nonnegative integers and all indices of these matrices start at (0, 0).

We write the matrices A and B which are the q-analagues of the matrices A and B, respectively. We give explicit expressions for LU-decompositions of these matrices and their inverses.

By help of a computer, LU-decompositions of these matrices were firstly ob-tained and then we have achieved the formulas by certain skills especially guessing skill. Using q-Zeilberger algorithm [5] and elementary matrix operations, the proofs are given as combinatorial identities. We will discuss a few of them here rather than all of them.

2. Decomposition of the Matrix A

The matrixA = [ˆaij] has the entries ˆaij =

[2i+m i ] q [2j+t j ]−1 q [i+j i ] q for 0≤ i, j ≤ n− 1. Now we will give expressions for LU-decompositions L1U1= A and L2U2=

A−1, for L−1₁ , U₁−1 and for L−1₂ , U₂−1 by the following theorem.

Theorem 2.1. For m,t≥ 0, (L1)ij = [ 2i + m i ] q [ i j ] q [ 2j + m j ]−1 q , ( L−1₁ )_ij = (−1)i−jq(i−j2 ) [ 2i + m i ] q [ i j ] q [ 2j + m j ]−1 q , (U1)_ij = qi 2₋₁[2i + m i ] q [ j i ] q [ 2j + t j ]−1 q , ( U₁−1)_ij = (−1)i−jq(j−i2 )−j 2₊₁[2i + t i ] q [ j i ] q [ 2j + m j ]₋₁ q ,

(L2)ij = (−1) i−j q(n−i−12 )−(n−j−12 ) [ 2i + t i ] q [ n− j − 1 i− j ] q [ 2j + 1 j ] q × [ i + j + 1 i ]−1 q [ 2j + t j ]−1 q , ( L−1₂ )_ij = q(n−i−1)(j−i) [ i + j j ] q [ n− j − 1 i− j ] q [ 2i + t i ] q [ 2i i ]−1 q [ 2j + t j ]−1 q , (U2)_ij = (−1)i−jq( n−j−1 2 )−(n−i−12 )−(n−i−1)(2i+1) [ 2i + t i ] q [ n + i i + j + 1 ] q , × [ i + j j− i ] q [ i + j j ]−1 q [ 2j + m j ]−1 q , ( U₂−1)_ij = q(n−j−1)(i+j+1) [ 2j + 1 j− i ] q [ i + j i ] q [ 2i + m i ] q [ 2j + t j ]₋₁ q × [ n + j i + j + 1 ]−1 q , and det A = n∏−1 k=0 qk2 [ 2k + m k ] q [ 2k + t k ]₋₁ q .

Proof. To prove L1L−11 = In where In is the identity matrix of order n, consider

∑ j≤k≤i (L1)_ik ( L−1₁ )_kj= (−1)j [ 2i + m i ] q [ 2j + m j ]−1 q ∑ j≤k≤i (−1)kq(k−j2 ) [ i k ] q [ k j ] q . Since _[ i k ] q [ k j ] q = [ i j ] q [ i− j k− j ] q , we have ∑ j≤k≤i (L1)ik ( L−1₁ )_kj = (−1)j [ 2i + m i ] q [ 2j + m j ]−1 q [ i j ] q ∑ j≤k≤i (−1)kq(k−j2 ) [ i− j k− j ] q = [ 2i + m i ] q [ 2j + m j ]−1 q [ i j ] q ∑ 0≤k≤i−j (−1)kq(k2) [ i− j k ] q .

Using Rothe’s formula, we see that (1; q)i−jis equal to 1 if i = j and 0 otherwise. Then we get _∑ j≤k≤i (L1)ik ( L−1₁ )_kj= δi,j,

as claimed, where δi,j is Kronecker delta. For U1 and U1−1, we write

∑ i≤k≤j (U1)_ik ( U₁−1)_kj = qi2 [ 2i + m i ] q [ 2j + m j ]₋₁ q ∑ i_≤k≤j (−1)k−jq(j−k2 )−j 2[k i ] q [ j k ] q = (−1)jqi2−j2 [ 2i + m i ] q [ 2j + m j ]₋₁ q [ j i ] q ∑ i_≤k≤j (−1)kq(j−k2 ) [ j− i k− i ] q = (−1)i+jqi2−j2+(j2)+( i+1 2 )−ij [ 2i + m i ] q [ 2j + m j ]−1 q [ j i ] q × ∑ 0_≤k≤j−i q(k+12 ) [ j− i k ] q ( −qi−j)k_.

By the Cauchy binomial theorem, for i̸= j, we get ∑ 0_≤k≤j−i q(k+12 ) [ j− i k ] q ( −qi_−j)k ₌ j−i ∏ k=1 (1− qi−j+k) = 0. Then _∑ i≤k≤j (U1)ik ( U₁−1)_kj= δi,j.

For LU-decomposition, we will show ∑ 0≤k≤min{i,j} (L1)_ik(U1)_kj= ˆaij, where A = [ˆaij] . Then ∑ 0≤k≤min{i,j} (L1)ik(U1)kj = [ 2i + m i ] q [ 2j + t j ]−1 q ∑ 0_{≤k≤min{i,j}} qk2−1 [ i k ] q [ j k ] q = q−1 [ 2i + m i ] q [ 2j + t j ]₋₁ q (q; q)_i(q; q)_j × ∑ 0_{≤k≤min{i,j}} qk2 1 (q; q)_i−k(q; q)_j−k(q; q)_k(q; q)_k.

Denote the last sum in the above equation by SUMk. The Mathematica version

of the q-Zeilberger algorithm [5] produces the recursion

SUMi = (1− qi+j) (1− qi₎2 SUMi−1. Since SUM0=_(q;q)1 i(q;q)j, we obtain SUMi = [ i + j i ] q 1 (q; q)_i(q; q)_j. Then we write _∑ 0_{≤k≤min{i,j}} (L1)ik(U1)kj= ˆaij.

For L2 and L−12 , we have

∑ j≤k≤i (L2)ik ( L−1₂ )_kj = (−1)iq(n−i−12 ) [ 2i + t i ] q [ 2j + t j ]−1 q × ∑ j_≤k≤i (−1)kq−(n−k−12 )+(n−k−1)(j−k) [ n− k − 1 i− k ] q [ 2k + 1 k ] q × [ i + k + 1 i ]−1 q [ k + j j ] q [ n− j − 1 k− j ] q [ 2k k ]−1 q = (−1)iq(n−1)(j+1)−(n2) [ 2i + t i ] q [ 2j + t j ]₋₁ q (q ; q)_i(q ; q)_n−j−1 (q ; q)_j(q ; q)_n−i−1 × ∑ j≤k≤i (−1)kq(k2)−kj (q ; q)_2k+1(q ; q)_k+j (q ; q)_i−k(q ; q)_i+k+1(q ; q)_k−j(q ; q)_2k.

By the q-Zeilberger algorithm for the second sum in the last equation, we obtain that it is equal to 0 provided that i̸= j. If i = j, it is obvious that (L2)ik

( L−1₂ )_kj= 1. Thus _∑ j≤k≤i (L2)ik ( L−1₂ ) kj= δi,j,

as claimed. Similarly we have ∑

i_≤k≤j

(U2)ik

(

For the LU−decomposition of A−1,we should thatA−1 = L2U2 which is same

as A = U₂−1L−1₂ . So it is sufficient to show that ∑ max{i,j}≤k≤n−1 ( U₂−1) ik ( L−1₂ ) kj = ˆaij. Hence ∑ max_{{i,j}≤k≤n−1} ( U₂−1)_ik(L−1₂ )_kj= [ 2i + m i ] q [ 2j + t j ]−1 q × ∑ max{i,j}≤k≤n−1 q(n−k−1)(i+k+1)+(n−k−1)(j−k) [ 2k + 1 k− i ] q [ i + k i ] q × [ n + k i + k + 1 ]₋₁ q [ k + j j ] q [ n− j − 1 k− j ] q [ 2k k ]₋₁ q = qn(i+j−1) [ 2i + m i ] q [ 2j + t j ]−1 q × ∑ max_{{i,j}≤k≤n−1} q(n−k−1)(i+j+1) [ 2k + 1 k− i ] q [ i + k i ] q × [ k + j j ] q [ n− j − 1 k− j ] q [ 2k k ]−1 q [ n + k i + k + 1 ]−1 q .

If we take (n + 1) instead of n, we write (2.1) ∑ j_≤k≤n q(n−k)(i+j+1) [ 2k + 1 k− i ] q [ i + k i ] q [ k + j j ] q [ n− j k− j ] q [ 2k k ]₋₁ q [ n + k + 1 i + k + 1 ]₋₁ q .

Denote sum in (2.1) by SUMn. For i̸= n and j ̸= n, the q-Zeilberger algorithm

gives the following recursion

SUMn= SUMn−1.

Thus, SUMn =SUMj =

[i+j i

]

q which completes the proof except the case (i, j) =

(n− 1, n − 1), which could be easily checked. The proof is obtained. 2

3. The Decomposition of the Matrix B

In this section, the matrixB = [

ˆ_bij]_{is defined with entries}

ˆ_{bij =} [ 2i + m i ]−1 q [ 2j + t j ] q [ i + j i ]−1 q

for 0≤ i, j ≤ n − 1. Now we will give expressions for LU-decompositions L3U3=B

and L4U4 =B−1, for L−13 , U3−1 and for L−14 , U4−1 without proof by the following

theorem

Theorem 3.1. For m, t≥ 0 and 0 ≤ i, j ≤ n − 1,

(L3)ij = [ 2j j ] q [ 2j + m j ] q [ i j ] q [ i + j j ]−1 q [ 2i + m i ]−1 q , ( L−1₃ )_ij = (−1)i−jq(i−j2 ) [ i + j− 1 j ] q [ 2j + m j ] q [ i j ] q [ 2i− 1 i− 1 ]₋₁ q [ 2i + m i ]₋₁ q , (U3)ij = (−1) i qi2+(i2) [ 2j + t j ] q [ i + j− 1 j− i ] q [ i + j i ]−1 q [ i + j− 1 j ]−1 q [ 2i + m i ]−1 q , ( U₃−1)_ij = (−1)iq(j−i2 )−( j 2)−j 2[j i ] q [ 2j + m j ] q [ 2j j ] q [ i + j− 1 i ] q [ 2i + t i ]₋₁ q , (L4)ij = (−1) i_−j q−(n−j−12 )+(n−i−12 ) [ n + i− 1 i ] q [ 2j + t j ] q [ n− j + 1 i− j ] q × [ 2i + t i ]−1 q [ n + j− 1 j ]−1 q , ( L−1₄ )_ij = q(n−i−1)(j−i) [ 2j + t j ] q [ n− j − 1 i− j ] q [ n + i− 1 i− j ] q [ 2i + t i ]₋₁ q [ i j ]₋₁ q , (U4)ij = (−1) n_−j−1 q(n−j−12 )+(i+n−1)(i−n+1) [ n + j− 1 i ] q [ n− i + j − 1 j ] q × [ n− i − 1 j− i ] q [ 2j + m j ] q [ 2i + t i ]₋₁ q . (U4)−1ij = (−1) j q(n2)−( j+1 2 )+i(n−j−1) [ 2j + t j ] q [ n− i − 1 j− i ] q [ n + i− j − 1 i ]−1 q × [ n + i− 1 j ]−1 q [ 2i + m i ]−1 q

and detB = n∏−1 k=0 (−1)kqk(3k−1)/2 [ 2k + t t ] q [ 2k + m k ]₋₁ q [ k + t k ]₋₁ q [ 2k− 1 k ]₋₁ q . 4. The Matrix A

In this section, we have the following results without proof by using the results of Theorem 2.1 with the fact given in (1.1). For 0≤ i, j < n − 1,

(L1)ij = ( 2i + m i )( i j )( 2j + m j )₋₁ , ( L−1₁ )_ij = (−1)i−j ( 2i + m i )( i j )( 2j + m j )₋₁ , (U1)ij = ( 2i + m i )( j i )( 2j + t j )₋₁ , ( U₁−1) ij = (−1) i_−j ( 2i + t i )( j i )( 2j + m j )−1 , (L2)ij = (−1) i_−j ( 2i + t i )( n− j − 1 i− j )( 2j + 1 j )( i + j + 1 i )−1(_{2j + t} j )−1 , ( L−1₂ )_ij = ( i + j j )( n− j − 1 i− j )( 2i + t i )( 2i i )−1(_{2j + t} j )−1 , (U2)ij = (−1) i_−j ( 2i + t i )( n + i i + j + 1 )( i + j j− i )( i + j j )−1(_{2j + m} j )−1 , ( U₂−1)_ij = ( 2j + 1 j− i )( i + j i )( 2i + m i )( 2j + t j )−1( _{n + j} i + j + 1 )−1 , det A = n∏−1 k=0 ( 2k + m k )( 2k + t k )₋₁ . 5. The Matrix B

In this section, we have the following results without proof by using the results of Theorem 2.2 with the fact given in (1.1). For 0≤ i, j < n − 1,

(L3)_ij = ( 2j j )( 2j + m j )( i j )( i + j j )₋₁( 2i + m i )₋₁ ,

( L−1₃ )_ij = (−1)i−j ( i + j− 1 j )( 2j + m j )( i j )( 2i− 1 i− 1 )−1(_{2i + m} i )−1 , (U3)_ij = (−1)i ( 2j + t j )( i + j− 1 j− i )( i + j i )−1(_{i + j− 1} j )−1(_{2i + m} i )−1 , ( U₃−1)_ij = (−1)i−1 ( j i )( 2j + m j )( 2j j )( i + j− 1 i )( 2i + t i )−1 , (L4)_ij = (−1)i−j ( n + i− 1 i )( 2j + t j )( n− j + 1 i− j )( 2i + t i )₋₁( n + j− 1 j )₋₁ , ( L−1₄ )_ij = ( 2j + t j )( n− j − 1 i− j )( n + i− 1 i− j )( 2i + t i )₋₁( i j )₋₁ , (U4)ij = (−1) n−j ( n + j− 1 i )( n− i + j − 1 j )( n− i − 1 j− i )( 2j + m j ) × ( 2i + t i )₋₁ , ( U₄−1)_ij = (−1)j ( 2j + t j )( n− i − 1 j− i )( n + i− j − 1 i )₋₁( n + i− 1 j )₋₁ × ( 2i + m i )₋₁ , det B = n∏−1 k=0 (−1)k ( 2k + t k )( 2k + m k )−1(_{k + t} k )−1(_{2k− 1} k )−1 .

References

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